On a generalized Appell system and monogenic power series

Abstract

Recently Appell systems of monogenic polynomials in R 3 were constructed by several authors. Main purpose of this paper is the description of another Appell system that is complete in the space of square integrable quaternion-valued functions. A new Taylor-type series expansion based on the Appell polynomials is presented, which can be related to the corresponding Fourier series analogously as in the complex one-dimensional case. These results find applications in the description of the hypercomplex derivative, the monogenic primitive of a monogenic function and the characterization of functions from the monogenic Dirichlet space.

A note on a one-parameter family of non-symmetric number triangles

“…We will try to omit as much as possible details from Clifford analysis, but due to the particular role in Clifford analysis we would like to stress in this introduction at least their origin as coefficients in the construction of generalized Appell polynomials in that framework. Those generalized Appell polynomials have recently received a lot of attention from several authors ( [3,8,15,16,20]) due to their important role in theory and applications ([6, 7, 9, 13]), specially in elasticity [3], PDE and special functions ( [5,12]), or 3D-quasiconformal mapping problems ( [10,11]). …”

On Generalized Hypercomplex Laguerre-Type Exponentials and Applications

“…It is also the reason why in the polynomial approximation in the context of Clifford Analysis almost every problem needs the development of different adapted polynomial bases (e.g. [1,3,4,5,11,18]). …”

On Paravector Valued Homogeneous Monogenic Polynomials with Binomial Expansion

“…If one uses, instead of the classical Appell sequence property F 0 (x) ≡ 1 from Definition 1.1, a generalized constant monogenic polynomial as an initial value like it was done in [2], then the set of P n k (x 0 , x) can also be extended and used in several applications, for instance in elasticity problems.…”

“…Since the construction of power-like monogenic functions was of general interest in Clifford analysis, the study of sets of Appell polynomials has developed meanwhile in several directions and has been realized with different methods and by different authors. We mention, for instance, [2] or [16]. Of particular interest remains to us the fact that in [16] a far reaching general representation theoretical method, based on Gelfand-Tsetlin bases, was developed.…”